A Closer Look: Scaling Analogies With Powers of Ten

It’s not easy to imagine dividing something into ten pieces nine
different times.

In order to make the difference in size scales more comprehensible,
we can expand the size of the original object to something with which
we are familiar. Let’s look at physicist Richard Feynman’s example
of expanding a hydrogen atom to the size of an apple.

To make this calculation
easier, we round off to the nearest power of ten. Feynman knew that an
atom is a few angstroms (10-10 meters) wide. Let’s
round off to 10 angstroms = 10-9 meters. An apple is about
10 centimeters (or 10-1 meters) wide. In order to make the
hydrogen atom as large as the apple, we have to make it ten times bigger
a total of eight times. In other words, we must expand it by a factor
of 108:

The size of hydrogen
atom (10-9 m) multiplied by the expansion factor (108)
equals the size of an apple (10-1 m)

or:
10-9 m x 108 = 10-1 m

An easy trick when
multiplying powers of ten is to simply add their exponents (the power
to which ten is raised). In the above example,
this shortcut
gives us

10-9 m x 108 = 10(-9 + 8) m
= 10-1 m

To rescale the size of the apple, so that
we may compare it to our expanded hydrogen atom, we multiply
the size of the
apple
by the
same expansion
factor:

The size of an apple (10-1) multiplied
by the expansion factor (108)
equals the size of earth (107 m or 10,000 km)
or:
10-1 m x 108 = 10(-1 + 8) =
107 m
(or 10,000 km)
The diameter of the earth is approximately 12,000 km, so
this is a good estimate.

Thus,
if you expand a hydrogen atom to the size of an apple, the apple would
expand to the size of the Earth. Having a student visualize the
difference
in scale between an apple and the Earth is more expressive than simply
stating that
an apple is 108 times bigger than an atom. This is the power
of these kinds of analogies.